A Study of Pedestrian Movement on Crosswalks Based on Chaos Theory

Document Type : Research Paper


1 M.Sc. Grad., Faculty of Engineering, Imam Khomeini International University, Qazvin,

2 Associate Professor, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran


Walking, as an important transportation mode, plays a large part in urban transportation systems. This mode is of great importance for planners and decision-makers because of its impact on environmental and health aspects of communities. However, this mode is so complex in nature that makes it difficult to study or model. On the other hand, chaos theory studies complex dynamical nonlinear systems that are sensitive to their initial conditions. A small change in initial conditions and/or parameters, may cause a big variation in the results. That is the situation that could happen in many fields of transportation. In the current study, the pedestrian behavior in crosswalks was studied in terms of chaos theory. The well-known social force model was chosen to model pedestrian movement in crosswalks, and based on the model, sensitivity analysis with respect to its parameters was carried out. Pedestrian road crossing behavior based on Helbing social force model was simulated in Matlab codes. Then pedestrian crossing behavior was investigated to detect the chaotic behavior. It was concluded that the speed of a pedestrian when the other pedestrians are closer than 100 cm and when the number of crossing pedestrians is more than 6 is chaotic. Moreover, increasing the number of pedestrians or decreasing the distance between pedestrians increase the occurrence of chaos. Chaotic behavior of speed causes turbulence in pedestrian crossing path, and that makes the path longer. Finally, some solutions for taking the system out of chaos, and consequently making its performance better, were proposed.


- Abshar, E. (2006) "Determining the necessity for investment in development of transportation network based on population capacity of the city: aplication of Chaos Theory",  (In Persian), M.Sc. Thesis, Sharif University of Technology.
- Adewumi, A., Kagamba, J. and Alochukwu, A. (2016) "Application of chaos theory in the prediction of motorised traffic flows on urban networks", Mathematical Problems in Engineering, Article ID 5656734, Vol. 2016, 15 pages.
- Barenghi, C. F. (2010) "Introduction to chaos: theoretical and numerical methods", CreateSpace Independent Publishing Platform.
- Chenga, A., Jiang, X., Li, Y., Zhangb, C. and Zhua, H. (2017) "Multiple sources and multiple measures based traffic flow prediction using the chaos theory and support vector regression method", Physica A, Vol. 466, pp. 422-434.
- Clifton, K. J., Singleton, P. A., Muhs, C. D. and Schneider, R. J. (2016) "Representing pedestrian activity in travel demand models: Framework and application", Journal of Transport Geography, Vol. 52,pp. 111-122.
- Eckmann, J.-P. and Ruelle, D. (1985) "Ergodic theory of chaos and strange attractors", Reviews of Modern Physics, Vol. 57, No. 3, pp. 617-656.
- Essex County Council (2006) "Designing for pedestrians: A guide to good practice", IHS BRE Press.
- Federal Highway Administration (FHWA) (2003) "Manual on uniform traffic control devices (MUTCD)", U.S.A.: Federal Highway Administration.
- Frazier, C. and Kockelman, K. M. (2004) "Chaos theory and transportation systems:an instructive example", Transportation Research Record: Journal of the Transportation Research Board, Vol. 1897, pp. 9-17.
- Gokyildirim, A., Uyaroglu, Y. and Pehlivan, I. (2016) "A novel chaotic attractor and its weak signal detection application", Optik - International Journal for Light and Electron Optics, Vol. 127, No. 19, pp. 7889-7895.
- Helbing, D. (1991) "A mathematical model for behavior of pedestrian",  Behavioral Science, Vol. 36, No. 4,pp. 298-310.
- Helbing, D., Farkas, I. J., Molnar, P. and Vicsek, T. (2002) "Simulation of pedestrian crowds in normal and evacuation situations", Pedestrian and evacuation dynamics.
- Helbing, D. and Molnar, P. (1995) "Social force model for pedestrian dynamics". Physical Review E, Vol. 51, No. 5.
- Helbing, D., Molnar, P., Farkas, I. J. and Bolay, K. (2001). "Self-organizing pedestrian movement", Environment and Planning B: Planning and Design, Vol. 28, pp. 361-383.
- Iran's Planning and Budget Organization (1996) "Walking Facilities" (In Persian) - Technical Deputy of Iran's Planning and Budget Organization.
- Jin Wanga, Q. S. (2013) "Short-term traffic speed forecasting hybrid model based on Chaos–Wavelet Analysis-Support Vector Machine theory", Transportation Research Part C: Emerging Technologies, Vol. 27,pp. 219-232.
- Kosiński, R. A. and Grabowski, A. (2011) "Langevin equations for pedestrian motion modeling". In:  American Conference on Applied Mathematics and The 5th WSEAS International Conference On Computer Engineering and Applications, January 29 - 31 2011 Puerto Morelos, Mexico. World Scientific and Engineering Academy and Society (WSEAS) 54-57.
- Krese, B. and Govekar, E. (2013) "Analysis of traffic dynamics on a ring road-based transportation network by means of 0–1 test for chaos and Lyapunov spectrum", Transportation Research Part C, Vol. 36,pp. 27-34.
- Kretz, T. (2015) "On oscillations in the social force model",  Physica A, Vol. 438, pp. 272-285.
- Lo, S.-C. and Cho, H.-J. (2005) "Chaos and control of discrete dynamic traffic model", Journal of the Franklin Institute, Vol. 342, No. 7,pp. 839-851.
- Lu, A. L., Ren, B. G., Wang, C. W. and Chan, D. C.-Y. (2015) "Application of SFCA pedestrian simulation model to the signalized crosswalk width design",  Transportation Research Part A, Vol. 80,pp. 76-89.
- Lu, L., Ren, G., Wang, W., Chan, C.-Y. and Wang, J. (2016) "A cellular automaton simulation model for pedestrian and vehicleinteraction behaviors at unsignalized mid-block crosswalks", Accident Analysis and Prevention, Vol. 95, No. Part B,pp. 425-437.
- Mahmoudabadi, A. (2014) "Assessment of applying chaos theory for daily traffic estimation", International Journal of Transportation Engineering, Vol. 1, No. 4, pp. 241-254.
- Mahmoudabadi, A. and Hosseini, S. M. S. (2013) "Application of chaos theory in hazardous material transportation", International Journal of Transportation Engineering, Vol. 1, No. 1, pp. 17-24.
- Narh, A. T., Thorpe, N., Bell, M. C. and Hill, G. A. (2016) "Do new sources of traffic data make the application of Chaos Theory to traffic management a realistic possibility?", Transport Reviews, Vol. 36, No. 5,pp. 635-658.
- Rafe, A. and Karimi, M. (2014) "Calibrating social force model based on design (of) experiments method", The 13th Tehran International Conference of Traffic and Transportation Engineering, 2014.
- Rahul, T. M. and Verma, A. (2017) "The influence of stratification bymotor-vehicle ownership on the impact of built environment factors in Indian cities", Journal of Transport Geography, Vol. 58, pp. 40-51.
- Rassafi, A. A., Poorzahedy, H. and Vaziri, M. (2006) "An alternative definition of sustainable development using stability and chaos theories", Sustainable Development, Vol. 14, No. 1,pp. 62-71.
- Seer, S., Rudloff, C., Matyus, T. and Brandle, N. (2014) "Validating social force based models with comprehensive real world motion data", Transportation Research Procedia, Vol. 2, pp. 724-732.
- Shahabi, M. (2010) "Network design for taking transportation network flows out of chaos: with and without uncertainty in supply (In Persian). M.Sc. Thesis, Sharif University of Technology.
- Shariat-Mohaimani, A., Klantari, N., Yazdanpanah, H. and Rafe, A. (2014) "Pedestrian Dynamic simulation in Tehran walking facilities (In Persian)", In:  The 13th Tehran International Conference of Traffic and Transportation Engineering, 2014.
- Sharifi, M. S., Shahabi, M., Abshar, E., Khorgami, M. H. and Poorzahedy, H. (2015) "Population capacity threats to urban area resiliency: observations on chaotic transportation network behavior", Scientia Iranica A, Vol. 23, No. 4,pp. 1675-1688.
- Skayannis, P., Goudas, M. and Rodakinias, P. (2017) "Sustainable mobility and physical activity: A meaningful marriage", Transportation Research Procedia, Vol. 24, No. C,pp. 81-88.
- Tiwari, G., Jain, D. and Rao, K. R. (2016) "Impact of public transport and non-motorized transport infrastructure on travel mode shares, energy, emissions and safety: Case of Indian cities", Transportation Research Part D, Vol. 44,pp. 277-291.
- Transportation and Traffic Deputy of Tehran Municipality (2014)  "Modeling and simulating of pedestrian walking in Tehran (In Persian), Tehran: Tehran Municipality.
- Transportation Research Board (2010)  "Highway capacity manual",  USA: Transportation Research Board, National Academies of Science.
- Van Zuylen, H., Geenhuizen, M. V. and Nijkamp, P. (1999) "(Un)predictability in traffic and transport decision making", Transportation Research Record: Journal of the Transportation Research Board, Vol. 1685, pp. 21-28.
- Wolf, A., Swift, J. B., Swinney, H. L. and Vastano, J. A. (1985) "Determining Lyapunov exponents from a time series", Physica D: Nonlinear Phenomena, Vol. 16, No. 3,pp. 285-317.
- Xu, M. and Gao, Z. (2009) "Chaos in a dynamic model of urban transportation network flow based on user equilibrium states", Chaos, Solitons and Fractals, Vol. 39, No. 2, pp. 586-598.
- Xu, M., Ye, Z. and Shan, X. (2015) "Modeling, analysis, and simulation of the co-development of road networks and vehicle ownership",  Physica A, Vol. 442, pp. 417-428.
- Xue, J. and Shi, Z. (2008) "Short-time traffic flow prediction based on chaos time series theory",  Journal of Transportation Systems Engineering and Information Technology, Vol. 8, No. 5,pp. 68-72.
- Yaling, F. and Zhongke, S. (2015) "Chaos analysis and delayed-feedback control in a discrete dynamic coupled map traffic model". Physica A, Vol. 422, pp. 40-46.
- Zębala, J., Ciępka, P. and Reza, A. (2012) "Pedestrian acceleration and speeds", Problems of Forensic Sciences, Vol. 91, pp. 227–234.
- Zeng, W., Chen, P., Nakamura, H. and Iryo-Asano, M. (2014) "Application of social force model to pedestrian behavior analysis at signalized crosswalk", Transportation Research part C, Vol. 40, pp. 143-159.
- Zeng, W., Chen, P., Yu, G. and Wang, Y. (2017) "Specification and calibration of a microscopic model for pedestrian dynamic simulation at signalized intersections: A hybrid approach", Transportation Research Part C, Vol. 80, pp. 37-70.
- Zeng, W., Nakamura, H. and Chen, P. (2014) "A modified social force model for pedestrian behavior simulation at signalized crosswalks", Procedia- Social and Behavior Sciences, Vol. 138, pp. 521-530.
- Zhang, X. and Jarrett, D. F. (1998) "Chaos in a dynamic model of traffic flows in an origin-destination network",  Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 8, No. 2, pp. 503-513.
- Zheng, Y., Chase, T., Elefteriadou, L., Schroeder, B. and Sisiopiku, V. P. (2015) "Modeling vehicle–pedestrian interactions outside of crosswalks". Simulation Modelling Practice and Theory, Vol. 59,pp. 89-101.